Evaluate The Expression:$\[ \frac{5}{6} + \frac{3}{8} - \frac{1}{10} \\]

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Introduction

When it comes to evaluating expressions involving fractions, it's essential to understand the rules of arithmetic operations and how to simplify fractions. In this article, we will delve into the world of fractions and explore how to evaluate the expression 56+38−110\frac{5}{6} + \frac{3}{8} - \frac{1}{10}.

Understanding the Basics of Fractions

Before we dive into the evaluation process, let's review the basics of fractions. A fraction is a way of expressing a part of a whole as a ratio of two numbers. It consists of a numerator (the number on top) and a denominator (the number on the bottom). For example, in the fraction 56\frac{5}{6}, the numerator is 5 and the denominator is 6.

Finding a Common Denominator

To add or subtract fractions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the fractions. In this case, the denominators are 6, 8, and 10. To find the LCM, we can list the multiples of each denominator:

  • Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
  • Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
  • Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100

The smallest number that appears in all three lists is 120, so the LCM of 6, 8, and 10 is 120.

Converting Fractions to Have a Common Denominator

Now that we have found the common denominator, we can convert each fraction to have a denominator of 120.

  • 56=5×206×20=100120\frac{5}{6} = \frac{5 \times 20}{6 \times 20} = \frac{100}{120}
  • 38=3×158×15=45120\frac{3}{8} = \frac{3 \times 15}{8 \times 15} = \frac{45}{120}
  • 110=1×1210×12=12120\frac{1}{10} = \frac{1 \times 12}{10 \times 12} = \frac{12}{120}

Adding and Subtracting Fractions

Now that we have converted each fraction to have a common denominator, we can add and subtract them.

  • 100120+45120=145120\frac{100}{120} + \frac{45}{120} = \frac{145}{120}
  • 145120−12120=133120\frac{145}{120} - \frac{12}{120} = \frac{133}{120}

Simplifying the Result

The final result is 133120\frac{133}{120}. However, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 133 and 120 is 1, so the fraction cannot be simplified further.

Conclusion

In conclusion, evaluating the expression 56+38−110\frac{5}{6} + \frac{3}{8} - \frac{1}{10} requires finding a common denominator, converting each fraction to have that denominator, adding and subtracting the fractions, and simplifying the result. By following these steps, we can simplify complex expressions involving fractions and arrive at a final answer.

Frequently Asked Questions

  • Q: What is the common denominator of 6, 8, and 10? A: The common denominator is 120.
  • Q: How do I convert a fraction to have a common denominator? A: To convert a fraction to have a common denominator, multiply the numerator and denominator by the necessary factor to make the denominator equal to the common denominator.
  • Q: How do I add and subtract fractions? A: To add and subtract fractions, find a common denominator, convert each fraction to have that denominator, and then add or subtract the numerators.

Final Answer

The final answer is 133120\boxed{\frac{133}{120}}.

Introduction

Evaluating expressions with fractions can be a challenging task, but with the right guidance, it can be made easier. In this article, we will address some of the most frequently asked questions related to evaluating expressions with fractions.

Q&A

Q: What is the common denominator of 6, 8, and 10?

A: The common denominator of 6, 8, and 10 is 120. To find the common denominator, list the multiples of each denominator and find the smallest number that appears in all three lists.

Q: How do I convert a fraction to have a common denominator?

A: To convert a fraction to have a common denominator, multiply the numerator and denominator by the necessary factor to make the denominator equal to the common denominator. For example, to convert 56\frac{5}{6} to have a denominator of 120, multiply the numerator and denominator by 20: 5×206×20=100120\frac{5 \times 20}{6 \times 20} = \frac{100}{120}.

Q: How do I add and subtract fractions?

A: To add and subtract fractions, find a common denominator, convert each fraction to have that denominator, and then add or subtract the numerators. For example, to add 56\frac{5}{6} and 38\frac{3}{8}, find the common denominator (120), convert each fraction to have that denominator, and then add the numerators: 100120+45120=145120\frac{100}{120} + \frac{45}{120} = \frac{145}{120}.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Q: How do I simplify a fraction?

A: To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). For example, to simplify 1218\frac{12}{18}, divide both the numerator and the denominator by 6: 12÷618÷6=23\frac{12 \div 6}{18 \div 6} = \frac{2}{3}.

Q: What is the least common multiple (LCM) of two numbers?

A: The LCM of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 6 and 8 is 24, because 24 is the smallest number that is a multiple of both 6 and 8.

Q: How do I find the LCM of multiple numbers?

A: To find the LCM of multiple numbers, list the multiples of each number and find the smallest number that appears in all lists. For example, to find the LCM of 6, 8, and 10, list the multiples of each number and find the smallest number that appears in all lists: 120.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the number on top of a fraction, and the denominator is the number on the bottom. For example, in the fraction 56\frac{5}{6}, the numerator is 5 and the denominator is 6.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator, then write the result as an improper fraction. For example, to convert 2 16\frac{1}{6} to an improper fraction, multiply the whole number by the denominator and add the numerator: 2×6+1=132 \times 6 + 1 = 13, then write the result as an improper fraction: 136\frac{13}{6}.

Conclusion

Evaluating expressions with fractions can be a challenging task, but with the right guidance, it can be made easier. By understanding the basics of fractions, finding a common denominator, converting fractions, adding and subtracting fractions, and simplifying the result, you can evaluate expressions with fractions with confidence. Additionally, by understanding the concepts of greatest common divisor, least common multiple, and mixed numbers, you can tackle even more complex problems.

Final Answer

The final answer is 133120\boxed{\frac{133}{120}}.